I am looking for a fast algorithm which enumerates 16-bit integers, sorted by the number of bits that are set.
this would necessarily be a two step process
1) convert entries to bitcount (I suggested a fast way)
2) sort

so, use any convenient sorting algorithm which will have something like
if (a>b)
and replace that by (add parantheses, typecasts and whatever)

if ((table[a >>8] + table[a & 0xff]) > (table[b >>8] + table[b & 0xff]))

Erik